Another look at binomial and related distributions exceeding values close to their centre

TL;DR

This paper extends known probability bounds for binomial distributions to include shifts from the mean, also applies similar results to Poisson and beta distributions, providing new insights into their behavior near the mode.

Contribution

It generalizes probability bounds for binomial distributions to shifted values, and demonstrates analogous properties for Poisson and beta distributions, linking these results.

Findings

Binomial probability bounds extend to shifted values.

Poisson and beta distributions exhibit similar properties.

The results provide deeper understanding of distribution behaviors near their modes.

Abstract

We generalise the known fact that for binomial $X_{n,k} \sim \mathrm{Bin}(n, k/n)$ one has $\inf_{k>1,n} \mathrm{P}(X_{n,k} \geq k) \geq \lim_{k \to 1+}\mathrm{P}(X_{2,k} \geq k) = 1/4$ to cover probabilities of exceeding a constant shift away from the mean. The proof is very short and the theorem yields the original result as a special case, as well as proving that an analogous result holds for the Poisson distribution. We furthermore prove a similar property holds for the family of beta distributions near their mode. Thanks to the connection between Binomial and Beta distributions, this allows us to shed some further light on the original result regarding Binomial probabilities.